The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  X  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 0 2X  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 2X 2X 2X 2X  0 2X  0  0 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X  0 2X  0 2X 2X 2X  0 2X 2X 2X  0 2X 2X 2X  0 2X 2X 2X  0  0  0 2X  0  0  0 2X  0  0  0 2X 2X  0 2X 2X
 0  0 2X  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X  0  0  0  0 2X 2X 2X 2X 2X  0 2X 2X  0 2X 2X  0 2X 2X  0 2X 2X  0  0 2X  0 2X 2X 2X  0 2X 2X 2X  0 2X 2X 2X  0 2X 2X 2X  0  0  0  0 2X  0  0 2X  0  0 2X 2X 2X  0 2X  0  0
 0  0  0 2X  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X 2X  0 2X 2X  0  0 2X 2X  0 2X 2X  0  0  0  0 2X  0  0 2X  0  0 2X 2X 2X  0 2X 2X  0  0 2X  0 2X  0  0 2X 2X  0  0 2X  0 2X 2X  0  0 2X 2X  0  0  0  0  0 2X 2X 2X 2X 2X  0  0 2X 2X  0 2X  0
 0  0  0  0 2X  0  0  0 2X 2X 2X 2X 2X  0 2X 2X  0  0  0  0  0  0  0  0  0  0  0  0  0 2X 2X 2X 2X 2X  0  0 2X 2X  0 2X 2X 2X  0 2X 2X  0 2X  0 2X  0 2X  0 2X 2X 2X  0 2X 2X 2X 2X  0 2X 2X 2X  0 2X  0  0 2X  0 2X 2X  0 2X  0 2X  0 2X  0  0  0  0
 0  0  0  0  0 2X  0 2X 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0  0 2X 2X 2X 2X  0  0  0 2X 2X  0 2X 2X  0 2X  0 2X  0  0 2X  0 2X  0  0 2X  0  0 2X 2X 2X  0 2X  0 2X  0 2X  0 2X  0  0 2X 2X  0  0 2X 2X  0 2X 2X  0  0 2X  0 2X 2X 2X  0  0  0 2X 2X  0
 0  0  0  0  0  0 2X 2X  0 2X 2X  0 2X 2X 2X  0  0 2X  0 2X 2X 2X  0  0 2X  0 2X 2X 2X  0  0 2X  0 2X  0  0  0 2X 2X 2X  0 2X 2X  0  0  0 2X  0 2X 2X 2X 2X 2X  0  0  0  0 2X  0  0  0 2X 2X 2X 2X  0  0 2X  0  0  0  0  0 2X  0 2X  0 2X  0 2X  0 2X

generates a code of length 82 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+32x^78+78x^80+832x^82+32x^84+32x^86+16x^88+1x^160

The gray image is a code over GF(2) with n=656, k=10 and d=312.
This code was found by Heurico 1.16 in 120 seconds.